To resonate at a certain frequency and match the output impedance of the transmitter (In this case around 27MHz and 50 Ohms) the antenna needs to be a certain length. The coils you see allow the physical length to be reduced but the electrical length to be maintained.
The physical length and diameter are up to the designer of the antenna.
If the designer wants the antenna to be able to handle lots of power the physical size must be adequate to handle the current.
In other words, if you are running a legal transmitter, the coil doesn't need to be the size of a Foster's pony keg.
Rule of thumb, the longer the receive area of the antenna, the more signal you will receive. A 108" stainless steel whip will handle all the power you can throw at it and provide the best reception area for the signals you are trying to receive. It is awfully noisy when you pull into the carport though.
If you are serious about building your own antennas may I suggest
ARRL Antenna Book as a great place to start. It will cover the theory and give plans and suggestions that are tried and true.
Bruce